If A={bmatrix} 0 & 1 \\ 1 & 0 {bmatrix}, then the value of A^{4} is

Correct Answer is : [1001]{bmatrix} 1 & 0 \\ 0 & 1 {bmatrix}[1001]

{bmatrix} 1 & 0 \\ 0 & 1 {bmatrix}

{bmatrix} 1 & 1 \\ 0 & 0 {bmatrix}

{bmatrix} 0 & 0 \\ 1 & 1 {bmatrix}

{bmatrix} 0 & 1 \\ 1 & 0 {bmatrix}

Correct Answer is : [1001]{bmatrix} 1 & 0 \\ 0 & 1 {bmatrix}[1001]

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UPSC NDA I 2017 Mathematics Solved Paper

Question : 41 of 120

Marks: +1, -0

A=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

A^{4}

Solution:

Concept:

Repeated multiplication of a matrix can simplify when it becomes the identity matrix.

Explanation:

Given

A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

First compute

A^2 = A \cdot A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

Top-left:

0\cdot0 + 1\cdot1 = 1

; top-right:

0\cdot1 + 1\cdot0 = 0

; bottom-left:

1\cdot0 + 0\cdot1 = 0

; bottom-right:

1\cdot1 + 0\cdot0 = 1

Thus

A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

, which is the identity matrix

I

Now

A^4 = (A^2)^2 = I \cdot I = I

Therefore

A^4 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Answer:

Option A:

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

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